Reduced Order Modeling using Principal Component Analysis
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Introduction
In this work model order reduction is proposed for the strong form Reproducing Kernel Collocation Method (RKCM), which is a meshfree numerical method based on points as discretization in the domain.
The main aim of model order reduction is to find a lower dimension approximation of a full model solution by projecting it onto a lower dimensional space. Proper Orthogonal Decomposition (POD), also known as Principal Component Analysis (PCA), is one of the most popular methods used to construct this projection operator. In this method the proper orthogonal modes of a system of equations are constructed and then these modes are truncated as required to construct the lower dimensional approximation. Model reduction for RKCM is done using a Least Squares Galerkin projection to project the over-determined system of equations. For the theory, full formulation and additional details of examples given below, refer to this pdf. The coding is done in MATLAB.
The main aim of model order reduction is to find a lower dimension approximation of a full model solution by projecting it onto a lower dimensional space. Proper Orthogonal Decomposition (POD), also known as Principal Component Analysis (PCA), is one of the most popular methods used to construct this projection operator. In this method the proper orthogonal modes of a system of equations are constructed and then these modes are truncated as required to construct the lower dimensional approximation. Model reduction for RKCM is done using a Least Squares Galerkin projection to project the over-determined system of equations. For the theory, full formulation and additional details of examples given below, refer to this pdf. The coding is done in MATLAB.
Numerical Examples
Examples from elasto-dynamics are given below which show the comparison of deformation between the full model and the reduced model using 'r' number of POD modes. For both examples, the end boundary conditions are fixed and an initial velocity is given. As time progresses, the deformations between the full and reduced solutions are compared.
Example 1: 1D Wave equation
In this simple example a single POD is enough to capture the deformation of the full scale solution.
Example 2: 1D Bi-material bar
In this example, the bar is made up of two materials in either half, which have different moduli of elasticity. More POD modes are required in order to obtain a reduced solution which captures the deformation of the full scale solution.